.Find the standard form of the equation of the parabola with focus(8,-2) and directrix x=4
well, you know that the parabola
y^2 = 4px
has
vertex at (0,0)
focus at x=p
directrix at x = -p
The vertex is halfway between the focus and the directrix, at (6,-2)
So, shift it right 6, down 2.
Then note that the distance from the focus to the directrix is 8-4=4=2p, and you have p=2. That means the equation is
(y+2)^2 = 8(x-6)
Confirm this at
http://www.wolframalpha.com/input/?i=parabola+(y%2B2)%5E2+%3D+8(x-6)
Why did the parabola go to therapy? Because it had focus issues with its directrix!
To find the standard form of the equation of the parabola with the given focus (8, -2) and directrix x = 4, we can use the definition of a parabola.
The standard form of the equation of a parabola is (x - h)^2 = 4p(y - k), where (h, k) is the vertex, and p is the distance between the vertex and the focus or directrix.
In this case, the vertex is the midpoint between the focus and the directrix. Let's find the vertex first.
The x-coordinate of the vertex is the average of the x-coordinates of the focus and the directrix:
x = (8 + 4) / 2
x = 12 / 2
x = 6
Since the directrix is a vertical line x = 4, the y-coordinate of the vertex remains the same as that of the focus: y = -2.
So, the vertex is (6, -2).
Now, we can find the distance between the focus and the vertex:
p = distance between the vertex and the focus
= |8 - 6|
= 2
Since p is positive, the parabola opens upwards.
Now, let's substitute the values into the standard form equation:
(x - 6)^2 = 4(2)(y + 2)
Simplifying, we have:
(x - 6)^2 = 8(y + 2)
Therefore, the standard form of the equation of the parabola with the given focus and directrix is (x - 6)^2 = 8(y + 2).
To find the standard form of the equation of a parabola with a given focus and directrix, we can use the definition of a parabola.
The definition states that for any point P on the parabola, the distance from P to the focus is equal to the perpendicular distance from P to the directrix.
Given the focus (8, -2) and the directrix x = 4, we need to find the equation of the parabola in the form (x-h)^2 = 4p(y-k), where (h, k) is the vertex and p is the distance between the vertex and either the focus or the directrix.
Step 1: Determine the vertex of the parabola.
The vertex of the parabola lies midway between the focus and the directrix. Since the directrix is a vertical line x = 4, the x-coordinate of the vertex is equal to the x-coordinate of the focus, which is h = 8. The y-coordinate of the vertex can be found by averaging the y-coordinates of the focus and the directrix, which gives us k = (-2+4)/2 = 1.
Therefore, the vertex of the parabola is (8, 1).
Step 2: Calculate the distance from the vertex to the focus or the directrix (p).
Since the directrix is a vertical line, the distance from the vertex to the directrix is the absolute value of the difference between the x-coordinate of the vertex and the x-coordinate of the directrix, which is |8-4| = 4.
Therefore, p = 4.
Step 3: Substitute the values into the standard form equation.
Using the values of h = 8, k = 1, and p = 4, we can substitute these into the standard form equation of a parabola.
(x - h)^2 = 4p(y - k)
(x - 8)^2 = 4(4)(y - 1)
(x - 8)^2 = 16(y - 1)
Therefore, the standard form equation of the parabola with focus (8, -2) and directrix x = 4 is (x - 8)^2 = 16(y - 1).